Double Wick rotating Green-Schwarz strings

HU-EP-14/65 HU-MATH-14/39 ZMP-HH-14/26

Prepared for submission to JHEP

Double Wick rotating Green-Schwarz strings

Gleb Arutyunov^a,1 and Stijn J. van Tongeren^b

aInstitut f¨ur Theoretische Physik, Universit¨at Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany

aZentrum f¨ur Mathematische Physik, Universit¨at Hamburg, Bundesstrasse 55, 20146 Hamburg, Germany

bInstitut f¨ur Mathematik und Institut f¨ur Physik, Humboldt-Universit¨at zu Berlin, IRIS Geb¨aude, Zum Grossen Windkanal 6, 12489 Berlin, Germany

E-mail: gleb.arutyunov@desy.de,svantongeren@physik.hu-berlin.de

Abstract:Via an appropriate field redefinition of the fermions, we find a set of conditions under which light cone gauge fixed world sheet theories of strings on two different back- grounds are related by a double Wick rotation. These conditions take the form of a set of transformation laws for the background fields, complementing a set of transformation laws for the metric and B field we found previously with a set for the dilaton and RR fields, and are compatible with the supergravity equations of motion. Our results prove that at least to second order in fermions, the AdS₅ ×S⁵ mirror model which plays an important role in the field of integrability in AdS/CFT, represents a string on ‘mirror AdS₅×S⁵’, the solution of supergravity that follows from our transformations. We discuss analogous solutions for AdS₃×S³×T⁴ and AdS₂×S²×T⁶. The main ingredient in our derivation is the light cone gauge fixed action for a string on an (almost) completely generic background, which we explicitly derive to second order in fermions.

1Correspondent fellow at Steklov Mathematical Institute, Moscow.

Contents

1 Introduction 1

2 Wick rotated bosons and changed geometry 3

3 Green-Schwarz fermions 5

4 Wick rotated fermions and changed background fields 6

4.1 Double Wick rotated Green-Schwarz fermions 7

4.2 The transformed Lagrangian 8

4.3 Fermions with metric and B field 10

4.4 Fermions with dilaton and RR fields 11

5 Double Wick related string backgrounds 12

6 Conclusions and outlook 16

A Appendices 18

A.1 Γ matrices,κ symmetry, and other conventions 18

A.2 Light cone gauge fixing 20

A.3 The spin connection 23

A.4 Type IIB supergravity 24

1 Introduction

A double Wick rotation on the world sheet of a string is a transformation that has proven its use in the context of the AdS/CFT correspondence [1]. Fixing a light cone gauge in the world sheet theory of a superstring on AdS₅×S⁵ [2] results in a model that transforms nontrivially under a double Wick rotation, and it is the thermodynamics of this different two dimensional quantum field theory [3, 4] that lies at the heart of the solution of the spectral problem in AdS/CFT using integrability [5, 6]. Namely, the spectrum of the string on AdS₅×S⁵ can be computed by means of the thermodynamic Bethe ansatz for this so-called mirror model [7–10].¹

We may wonder whether such double Wick rotated models have a more direct physical interpretation. In previous work we showed that (given mild restrictions) doing a double Wick rotation at the bosonic level is equivalent to a particular transformation of the metric and B field [14]. Here we will establish the analogous result for the fermions, and find the

1These equations have been used to get impressive finite coupling results (see e.g. [11]), and have been simplified to the so-called quantum spectral curve [12], particularly impressive in the perturbative regime (see e.g. [13]).

transformation of (a combination of) the dilaton and Ramond-Ramond (RR) background fields such that the complete transformation is equivalent to a double Wick rotation of the Green-Schwarz action (to second order in fermions).

In order to find this transformation, we take a type IIB Green-Schwarz string to second order in fermions on an (almost) completely generic background with light cone isometries, and fix a light cone gauge. We then specify our restrictions on the metric and B field and get the set of fermionic terms that are to be analytically continued. As fermions have single derivative kinetic terms, their double Wick rotation is subtle and requires a complex field redefinition; fortunately this can be unambiguously fixed in the flat space limit. We then proceed by attempting to ‘undo’ the double Wick rotation, knowing the transformation that accomplishes this at the bosonic level. Hence we take the general double Wick rotated action and rewrite it in terms of the transformed metric and B field, changing the light cone block of the metric as glc → glc/|detglc| and the sign of the (transverse) B field.

As an immediate consistency check we find that all fermionic terms fixed by the metric and B field directly match the original action as they must. We then compare the double Wick rotated dilaton–RR terms (with transformed metric and B field) to the originals and read off how they should transform. For our considerations to apply straightforwardly, we find that similarly to how we assume our metric and B field not to mix light cone and transverse directions, the RR fields should not lead to such mixing either. This translates to the requirement that the RR fields should always have a single light cone index. The dilaton and allowed RR fields then need to transform such that the combinationse^ΦF_a(bcde)^(1/5) are invariant, whilee^ΦF_abc⁽³⁾picks up a sign, likedB. Of course these RR fields now also live in a different space, so that the unchanging nature of e^ΦF is only apparent. While we do not work out the details, the same analysis can be readily repeated for the type IIA string.

Then, by tracing the world sheet double Wick rotation back through the light cone gauge fixing procedure,² we discuss how to interpret the transformation of the background fields from a target space perspective, as a combination of T dualities and target space analytic continuation (Wick rotation). Though the analytic continuation is somewhat subtle, this shows that our transformation is compatible with the supergravity equations of motion, precisely under our restrictions on the background fields. Our main example is AdS₅×S⁵ and its cousin ‘mirror AdS₅×S⁵’, whose supergravity solution we constructed in [14]. The associated background fields match our transformations, thereby explicitly proving (to second order in the fermions) that the AdS₅ ×S⁵ mirror model physically represents a string on mirror AdS₅×S⁵. We similarly briefly discuss mirror solutions for AdS₃ ×S³×T⁴ and AdS₂ ×S² ×T⁶. While our restrictions on the background do not appear too drastic, spaces such as e.g. AdS₄×CP³ do not fit them. In the outlook we discuss the apparent obstacles that arise when we relax our restrictions.

We should note that our work was ultimately inspired by the ‘mirror duality’ [15]

of the integrable deformation of the AdS₅ ×S⁵ coset model of [16, 17].³ Labeling the deformation by a parameterκ, an exact S-matrix approach to this model showed that the

2We would like to thank A. Tseytlin for this suggestion

3For further developments in this area see e.g. [18–30].

light cone theory at 1/κis equivalent to the double Wick rotation of the theory at κ [15].

This statement has a simple proof at the bosonic level using the metric and B field found in [18], and in the κ → 0 limit gives precisely (mirror) AdS₅ ×S⁵ [14]. The results of our paper should allow us to check this statement at the fermionic level, assuming this model corresponds to a string sigma model. However, to date the dilaton and RR fields of this model have not been found, and whether this and related models represent strings for generic values ofκ is an interesting open question. We will briefly come back to this in our conclusions.

In the next section we will begin our discussion by summarizing the transformation of the metric and B field that is equivalent to a double Wick rotation at the bosonic level. In section3 we introduce fermions on the world sheet, and present the light cone gauge fixed action derived in appendixA.2. We then discuss the double Wick rotation of fermions, the matching of the metric and B field terms, and the transformations of and constraints on the dilaton and RR fields in section4. Section5contains the discussion of our transformation from a target space perspective, along with the examples of AdS₅ ×S⁵, AdS₃ ×S³ ×T⁴, and AdS₂×S²×T⁶ and their mirror partners. We finish in section 6 by discussing open questions and generalizations of our work.

2 Wick rotated bosons and changed geometry

To set the stage, let us recall the change of geometry that is equivalent to doing a double Wick rotation on the world sheet of a bosonic light cone string [14].

The action describing bosonic string propagation on a generic background is given by S≡ −^T₂

Z

dτdσL=−^T₂ Z

dτdσ (g_mndx^mdxⁿ −B_mndx^m∧dxⁿ) ,

whereT is the (effective) string tension. We will considerd-dimensional backgrounds with coordinates{t, φ, x^µ}, packaging tand φin the (generalized) light cone combinations

x⁺= (1−a)t+aφ, and x⁻=φ−t , (2.1) where a is a free parameter.⁴ Further conventions are summarized in appendix A.1. We take the metric to be of the block diagonal form

g_mn =







g₊₊ g+− 0 g+− g−− 0 0 0 gµν





≡ g_lc 0 0 gµν

!

, (2.2)

with components depending only on the transverse coordinates x^µ, and consider purely transverse B fields. We will address the question of relaxing these restrictions in section6.

As discussed in more detail in appendix A.2, to fix a uniform light cone gauge we impose⁵

x⁺ =τ , p−= 1. (2.3)

4At the level of gauge fixing, this parameter interpolates between the temporal gauge ata= 0 and the conventional light cone gauge ata= 1/2 [31]. In [14] we tooka= 0 for conciseness.

5We focus on the zero winding sector in cases whereφparametrizes a circle the string can wind around.

L

⁽⁰⁾

L ˜

⁽⁰⁾

τ→i˜σ, σ→ −i˜τ

glc→ _|det^glc_g

lc|, B→ −B

Figure 1. A double Wick rotation as a change of geometry. At the bosonic level a double Wick rotation of the light cone world sheet theory of a string is equivalent to a change of the metric and B field.

The resulting action takes the form S⁽⁰⁾ =−T

Z

dτdσ ^pY⁽⁰⁾+g⁺⁻/g⁺⁺−x˙^µx^0νB_µν, (2.4) where

Y⁽⁰⁾= ( ˙x^µx⁰_µ)²− AC g⁺⁺g−−

(2.5) with

A= 1 +g−−x^0µx⁰_µ,

C = 1 +g⁺⁺x˙^νx˙ν, (2.6)

and dots and primes refer to derivatives with respect to time (τ) respectively space (σ).

Note that we work in units that put spatial and temporal derivatives on the same footing.

Now we readily see that a double Wick rotation of the world sheet coordinates

τ →i˜σ, σ → −i˜τ , (2.7) gives an action of the same form, with g−− exchanged for −g⁺⁺ while leaving g⁺⁻/g⁺⁺

unchanged,⁶ and with a change of sign on B. This change of the metric is nothing but g_lc→g⁻¹_lc = g_lc

|detglc|, (2.8)

which applies in any coordinate system in the light cone subspace. In particular it is simply gtt ↔ g^φφ when the original metric is diagonal in t and φ.⁷ Put differently, the action is formally left invariant under a double Wick rotation combined with the transformations (2.8) and B → −B on the background. We have illustrated this idea in figure1. Keep in mind that this combined transformation interchanges Aand C and leaves Y⁽⁰⁾ invariant.

At this stage it is not clear that the mirror space associated to a string background represents a string background itself. We will come back to this point in detail later, for now let us simply assume there are examples where this is the case. We would then like to see that the fermions of such strings behave appropriately under this transformation as well. Let us therefore add fermions to our string, and fix a light cone gauge again, so that we can determine the required transformation properties of the dilaton and the RR fields.

6Sinceg⁺⁻/g⁺⁺=−g+−/g−−, this is equivalent to exchangingg⁺⁻forg+−.

7We defineds²=−gttdt² (+2gtφdtdφ) +gφφdφ²+gµνdx^µdx^ν.

3 Green-Schwarz fermions

As we are interested in string backgrounds with RR fields, we need to use the Green- Schwarz formalism. Though a fully explicit construction is not known, the Lagrangian describing propagation of a Green-Schwarz superstring in an arbitrary background can be constructed second order by second order in the fermions, explicitly done up to fourth order in [32].⁸ We will limit ourselves to second order, which will provide a nontrivial test of our ideas. As AdS₅×S⁵ is one of our spaces of interest, we will focus on the type IIB string. The Lagrangian (density) at quadratic order in the fermions is given by [32]

L⁽²⁾_f =i?e^aθΓ¯ aDθ−e^aθΓ¯ aσ₃Dθ , (3.1) where θ= (θ₁, θ₂) is a doublet of sixteen component Majorana-Weyl spinors, withσ₃ and the≡iσ2 term inD (see below) acting in this two dimensional space. Dis given by

D=d−¹₄ω/ +¹₈e^aH_abcΓ^bcσ₃+¹₈e^bSΓ_b, (3.2) where ω is the spin connection, e the (one form) vielbein, and H = dB is the Neveu- Schwarz–Neveu-Schwarz (NSNS) three form. Slashes denote contraction with the appro- priate product of Γ matrices, Γ_a...m. S contains the dilaton and RR fields

S =−e^Φ( /F⁽¹⁾+_3!¹σ₁F/⁽³⁾+_2·5!¹ /F⁽⁵⁾), (3.3) with theF⁽ⁿ⁾ denoting thenform RR fields, and Φ the dilaton. In components this reads L⁽²⁾_f =iγ^αβ∂αx^mθΓ¯ _mD_βθ−i^αβ∂αx^mθΓ¯ _mσ₃D_βθ , (3.4) with

D_β =∂β−¹₄ω/_m∂βx^m+¹₈∂βx^mH_mnpΓ^npσ3+¹₈SΓ_m∂βx^m. (3.5) We prefer to write out the Lagrangian as

L=γ^αβˆg_mn∂_αx^m∂_βxⁿ −^αβBˆ_mn∂_αx^m∂_βxⁿ

+iγ^αβ∂_αx^mθΓ¯ _m∂_βθ−i^αβ∂_αx^mθΓ¯ _mσ₃∂_βθ , (3.6) where the hats indicate these terms are not purely the bosonic metric and B field, i.e.

ˆg_mn =g_mn− ₄ⁱθΓ¯ _(mω/_n)θ+₈ⁱθΓ¯ _(mH_n)pqΓ^pqσ3θ+₈ⁱθΓ¯ _(mSΓ_n)θ, Bˆ_mn =B_mn −₄ⁱθΓ¯ _[mω/_n]σ3θ+₈ⁱθΓ¯ _[mH_n]pqΓ^pqθ+₈ⁱθσ¯ 3Γ_[mSΓ_n]θ,

(3.7) with round and rectangular brackets denoting symmetrization and antisymmetrization re- spectively, defined with the usual factor of 1/n!. In this notation we will treat ˆg as if it really were the metric, so it can be used to raise and lower indices at intermediate stages of computation.⁹ When light cone gauge fixing, the fermions that contribute to terms in

8To second order this Lagrangian was first derived in [33,34].

9The price of this hopefully unambiguous notation is having to keep track of extra signs in fermionic terms in the inverse ‘metric’. Note that while not explicitly done in this paper, this notation is very useful for example when fixing a light cone gauge in a Hamiltonian setting.

gˆand ˆB that are already nonzero at the bosonic level formally go along for the (bosonic) ride, while possible extra nonzero terms need only be kept at a linearized level (quadratic in fermions). Of course we still need to add the manifestly fermionic terms in eq. (3.6) to the derivation. The only assumption we will make at this stage is that the fermions do not generate a nonzero ˆB+−, which will turn out to follow from our restrictions on the metric and B field and the restrictions on the RR fields we will find later.

Light cone action

In appendixA.2we fix a light cone gauge for a string on a completely generic background with light cone isometries andB+− = 0, including fermions to second order. For the type of backgrounds we are considering, this gives the gauge fixed action

S=S^(0,2)−T Z

dτdσL⁽²⁾_a +L⁽²⁾_b , (3.8) whereS^(0,2) is the bosonic gauge fixed action (2.4) with metric and B field replaced by (the relevant parts of) ˆg and ˆB. L⁽²⁾a and L⁽²⁾_b are given by

L⁽²⁾_a = 1 2

√ Y⁽⁰⁾

1 g−−g⁺⁺

h

(2ˆg^+µx˙µ−iθΓ¯ ⁺θ)A˙ + (2 ˆB−µx^0µ+iθΓ¯ −σ3θ⁰)C (3.9)

−g−−(2ˆg^+µx⁰_µ−iθΓ¯ ⁺θ⁰) +g⁺⁺(2 ˆB−µx˙^µ+iθΓ¯ −σ₃θ)˙ x˙_µx^0µi and

L⁽²⁾_b = − 1 g−−

i 2

θΓ¯ −θ˙+ ˆg−µx˙^µ

− 1 g⁺⁺

i 2

θΓ¯ ⁺σ₃θ⁰+ ˆB^+µx⁰_µ

(3.10) which are the formally new terms introduced by the fermions. A and C are those of eqs.

(2.6). In this action we have fixed a κ symmetry gauge by taking

(Γ⁰+ Γ^p)θ= 0, (3.11)

where Γ⁰ and Γ^p are the flat space cousins of Γ^t and Γ^φ, see appendixA.1. This means we essentially dropped terms of the form ¯θΓ^µθ in Γ matrix structure; appendix A.2 contains the full action before imposing aκ symmetry gauge choice.

4 Wick rotated fermions and changed background fields

In section2we recalled how at the bosonic level a double Wick rotation on the world sheet is actually equivalent to a change of background. We would now like to investigate this at the fermionic level. To do so, we will proceed as before, carefully considering a double Wick rotation of the general action (3.8), and attempting to reinterpret the result as an action of the same form.

The story is slightly more involved for fermions than for bosons, due to their single derivative kinetic terms. Already for the massless free Dirac Lagrangian it is clear that a double Wick rotation does not result in a real Lagrangian if we keep the conjugation properties of the fermions fixed. Hence we need to accompany a double Wick rotation by a

L L ˜ L ˇ

τ →i˜σ, σ→ −i˜τ

glc→ _|det^glc_g

lc|

B→ −B S →S˜

Figure 2. A double Wick rotation as a change of background fields? Combining a double Wick rotation with a change of the metric and B field and insisting the result agrees with the original action should fix the transformationS →S˜of the dilaton and RR fields. The intermediate ˇLhas no obvious physical interpretation.

change of reality condition, or equivalently, a complex field redefinition.¹⁰ We will fix this field redefinition by considering a Green-Schwarz string in flat space, which we will then use to find the double Wick rotated action on a general background.

To reinterpret the result as an action of the form (3.8) again, we will attempt to com- plete the bosonic diagram of figure1to the one of figure2, finding the total transformation of the background fields that ‘undoes’ the double Wick rotation.

4.1 Double Wick rotated Green-Schwarz fermions

To determine the precise transformation of our fermions under a double Wick rotation, we will consider a Green-Schwarz string in flat space. The light cone Lagrangian for such a string is given by¹¹

L= T 2

x˙^µx˙µ−x^0µx⁰_µ+iθΓ¯ ⁻θ˙+iθΓ¯ ⁻σ3θ⁰. (4.1) Now let us carefully do a double Wick rotation

x(τ, σ)→x(i˜σ,−i˜τ)≡y(˜τ ,σ)˜

θ(τ, σ)→θ(i˜σ,−i˜τ)≡θ(˜˜τ ,σ)˜ (4.2) so that we have

x⁰ =iy,˙ x˙ =−iy⁰, θ⁰ =iθ,˙˜ θ˙=−iθ˜⁰,

(4.3) where dots and primes refer to derivatives in the first and second arguments. The resulting action then takes the form

L= T 2

y˙^µy˙_µ−y^0µy⁰_µ+θΓ¯˜ ⁻θ˜⁰−θΓ¯˜ ⁻σ₃θ˙˜. (4.4)

10In the context of the AdS5×S⁵ coset sigma model this was noted explicitly in [4], here we will be considering a transformation of the canonical Green-Schwarz fermions in a general setting.

11This can be obtained from our general action in flat space, taking the limita→1/2 from below. Note that in flat space witha= 1/2, Γ⁺=G⁺, whereG⁺ is the matrix involved inκsymmetry gauge fixing as discussed in appendixA.1.

We see that the bosonic part of this Lagrangian is formally the same as that of the original Lagrangian (4.1), and we can assume y to have the reality properties of x; typically they are not even distinguished. If we however assume ˜θ(˜τ ,˜σ) to have the reality properties of θ(τ, σ), this action will not be real when the original Lagrangian is. To fix this we should change the reality properties of ˜θ. Equivalently, we would like to consider a (constant, complex) field redefinition

θ˜=M η, (4.5)

where the action is real when written in terms ofη with conventional reality properties. A priori it is not clear how to fixM exactly, but we have an additional physical requirement to impose.

As a light cone string in flat space is a Lorentz invariant model (manifest in the NSR description), a double Wick rotation should leave all physical properties of the model invariant. The simplest way to realize this is to insist that the form of the Lagrangian is invariant under a double Wick rotation. Comparing eqs. (4.1) and (4.4), we see that this is the case if

ηM¯ ^tΓ⁻M η⁰ =i¯ηΓ⁻σ₃η⁰, (4.6) and

ηM¯ ^tΓ⁻σ3Mη˙=−i¯ηΓ⁻η.˙ (4.7) Not having or wanting to touch the Γ matrices we need

M^tM =iσ3 M^tσ3M =−i1. (4.8)

This fixes

M = 0 b c 0

!

, (4.9)

withc² =−b²=i, mixing the two Majorana-Weyl spinors. The overal sign ofM is clearly inconsequential, but this still leaves the choice of bc = ±1. Later we will see that this choice is actually inconsequential as well. In summary, a double Wick rotation can be implemented by the replacements

x⁰ →ix,˙ x˙ → −ix⁰,

θ→M η, θ⁰ →iMη,˙ θ˙→ −iM η⁰, (4.10) where we have stopped carefully distinguishing the bosons.

4.2 The transformed Lagrangian

Looking at figure2it is clear that we are primarily interested in ˇL, the result of transforming L by combining a double Wick rotation with a change of the metric and B field so that the bosonic part of the action is left invariant. This change comes with strings attached however, since it appears hard to give a target (or flat) space interpretation to the result of transforming the metric in a target space Γ matrix; for a diagonal metric, Γ^t=^pg^ttΓ⁰

becomes √

g_φφΓ⁰ for example. This can be fixed by the automorphism of our Clifford algebra taking

Γ^0/p→iΓ^p/0, (4.11)

where p is the other flat light cone direction, associated to φ. This automorphism is compatible with ourκsymmetry gauge fixing, and is equivalent to a unitary change of basis by e^{−iπ4Γ0Γp} on the fermions.¹² Combining the change of metric with this automorphism we get¹³

Γ^± Γ±

!

→ Γˇ^± Γˇ±

!

→ ±i Γ_∓

−Γ^∓

!

, (4.12)

where the rightmost Γ matrices now have proper target space indices with respect to the changed metric. The factors of i will disappear again since thanks to our κ symmetry gauge we also have

η¯→i¯η. (4.13)

Taking the action (3.8) and doing the above double Wick rotation, changing the metric and B field and implementing this automorphism we get

S= ˇS^(0,2)−T Z

d˜τd˜σLˇ⁽²⁾_a + ˇL⁽²⁾_b , (4.14) with

Lˇ⁽²⁾_a = 1 2

√ Y⁽⁰⁾

1 g−−g⁺⁺

h(i2Bˇˆ−µx˙^µ−i¯ηΓ⁺η)A˙ + (−i2ˇˆg^+µx⁰_µ+i¯ηΓ−σ₃η⁰)C (4.15)

−

g−−(i2Bˇˆ−µx^0µ−i¯ηΓ⁺η⁰) +g⁺⁺(−i2ˇˆg^+µx˙_µ+i¯ηΓ−σ₃η)˙

x˙_µx^0µi and

Lˇ⁽²⁾_b = − 1 g−−

i

2ηΓ¯ −η˙−iBˇˆ^+µx˙µ

− 1 g⁺⁺

i

2ηΓ¯ ⁺σ₃η⁰+igˇˆ−µx^0µ

. (4.16)

Remaining checks denote transformed quantities we have temporarily left implicit. Com- paring this to the general action (3.8), we see that the manifestly fermionic terms match perfectly withη=θ. From the remaining terms we read off that we need to have

ˇˆ

g^+µ=iBˆ₋^µ, Bˇˆ₋^µ=−iˆg^+µ, ˇˆ

g−µ=−iBˆ⁺_µ, Bˇˆ⁺_µ=iˆg−µ.

(4.17) We also have to check that the fermionic terms in ˇS^(0,2) behave as the metric and B field, i.e. gˇˆ−−=−ˆg⁺⁺, gˇˆ⁺⁺=−ˆg−−,

ˇˆ

g+−= ˆg⁺⁻, gˇˆ⁺⁻= ˆg+−, ˇˆ

g_µν = ˆg_µν, Bˇˆ_µν =−Bˆ_µν.

(4.18)

12Of course, in ourκsymmetry gauge this is just multiplication bye^−iπ4, and actually does nothing to our expressions. This is not a particularly useful point of view however.

13Restricting ourselves to the 2d light cone block, starting with the original vielbeinesatisfyinge^tηe=g we can construct a ‘mirror’ vielbein as ˜e=g⁻¹eσ1, since the mirror metric is justg⁻¹. This means we haveg⁻¹eηΓ =e σ˜ 1ηΓ, i.e. e^±_aΓ^a=∓˜e^a∓(σ1)_a^bηbcΓ^c. Combined with the transformation Γ→iσ1Γ we get the desired result.

Finally, in order not to leave our present framework, the assumption ˆB+−= 0 needs to be preserved. We will begin by checking that the fermionic terms in ˆg and ˆB determined by the metric and B field actually behave as they should.

4.3 Fermions with metric and B field

The fermionic terms in ˆg and ˆB that have only the metric or B field in them are ¯θΓ_mω/

nθ and ¯θΓ_mH_npqΓ^pqσ3θ, with an extra σ3 inserted for ˆB. Our κ symmetry gauge choice together with our restrictions on the metric and B field imply that many of these terms vanish. Using the block notation of eq. (2.2) we have

θΓ¯ _mω/_nθ= 0 •

• 0

mn

, (4.19)

where bullet points denote asymmetric (generically) nonzero contributions. In other words the spin connection term in ˆgand ˆBonly contributes to ˆg±µand ˆB±µ, which by assumption have no bosonic term. H contributes similarly

θΓ¯ _mH_npqΓ^pqθ= 0 •

• 0

mn

. (4.20)

These terms therefore only contribute to the relations (4.17).

The first of eqs. (4.17) requires the spin connection terms to transform as

θΓ¯ ⁽⁺ω/^µ)θ→ −iθΓ¯ _[−ω/^µ]σ₃θ (4.21) under the full set of transformations used to arrive at ˇL, since the fermionic contribution to the inverse of the ‘metric’ ˆg has raised indices but an opposite sign. Now under these transformations the light cone components of the spin connection change as

/ ω^± ω/_±

!

→ ∓i ω/_∓

−ω/^∓

!

, (4.22)

while ω/_µ is unaffected, see appendixA.3 for details. Combining this with all other trans- formations we get

θ¯Γ⁺ω/^µ+ Γ^µω/⁺θ→(i¯η)(iΓ−)ω/^µ+ Γ^µ(−i/ω₋)(iσ₃)η=−i¯ηΓ₋ω/^µ−Γ^µω/₋σ₃η, nicely matching eq. (4.21) upon identifying η = θ. The rest of eqs. (4.17) works analo- gously, the relative sign arising cf. eqs. (4.8).

TheHcontribution to these relations has an extraσ₃ but otherwise proceeds similarly.

The last of eqs. (4.17) for example requires

θΓ¯ ^[+H_µ]νρΓ^νρθ→iθΓ¯ ₍₋H_µ)νρΓ^νρσ₃θ, while our transformation gives

θΓ¯ ⁺H_µνρΓ^νρθ→(i¯η)(iΓ−)(−H_µνρ)Γ^νρ(iσ₃)η=i¯ηΓ−H_µνρΓ^νρσ₃η matching precisely sinceH is purely transverse.

4.4 Fermions with dilaton and RR fields

Having checked that the metric and B field terms behave consistently, let us investigate the fermionic terms involving the dilaton and RR fields. A priori these terms can give contributions to any part of ˆg and ˆB. If our transformation is to work we get immediate restrictions however, due to the transformation properties of the spinor structure inS (see eq. (3.3))

M^tM=−bc, M^tσ1M =bcσ1, (4.23)

actually leaving the type of couplings invariant. Moreover these transformations guarantee that a realiθΓ¯ _MSΓ_nθremains real, sincebc=±1 and fermionic terms that are nonzero in ourκgauge contain an even number of non-transverse Γ matrices. This makes it impossible for the contribution ofS to relate the double Wick rotation of ˆg, to ˆB, and vice versa.¹⁴

From the fact that our forms cannot lead to a mixing of light cone and transverse components in ˆg and ˆB,

θΓ¯ ^±SΓ^µθ= ¯θΓ^µSΓ^±θ= 0, (4.24) we deduce that each nonzero term in S must have a single light cone index, since three or more light cone indices necessarily give zero, and two or none gives a nonzero contribution in the above. Note that these conditions also immediately imply ˆB+− = 0. We then have to insist these terms transform to match eqs. (4.18). For the light-cone components we

have θΓ¯ ⁺SΓ^±θ

θΓ¯ ⁻SΓ^±θ

!

→ ±i −θΓ¯ −SΓˇ _∓θ θΓ¯ ₊SΓˇ ∓θ

!

, (4.25)

where

Sˇ=−bcS

_Γ^0/5_→iΓ^5/0

σ1→−σ1

. (4.26)

The purely transverse components simply give

θΓ¯ _µSΓ_νθ→iθΓ¯ _µSˇΓ_νθ. (4.27) In the ˆB case we get an extra sign here due to σ3. Taking into account the extra minus sign for the fermions in the inverse of ˆg, these transformations are compatible with eqs.

(4.18) provided we have a set of mirror fields (packaged in ˜S) such that θΓ¯ _mSΓ˜ _nθ=







iθΓ¯ _mSΓˇ _nθ m,n∈ {µ},

−iθΓ¯ _mSΓˇ _nθ m,n∈ {+,−}. (4.28) The problematic looking relative sign we can now remove by inserting 1 in the form of (Γ⁰Γ^p)², giving

θΓ¯ _mSΓ˜ _nθ=−iθΓ¯ _mΓ⁰Γ^pSΓˇ _nθ (4.29)

14Since σ1σ3 =−and in our κsymmetry gauge we have Γ⁰Γ^pθ =θ, we might imagine that (the 0p containing components of) an F⁽ⁿ⁾ term effectively starts looking like an F⁽ⁿ⁻²⁾ term with an extraσ3. However these terms would not couple correctly to the bosons (sitting in ˆgrather than ˆB or vice versa), and moreover would still not pick up the required factor of i. Still, we could more specifically imagine a pair of forms such thatF/⁽ⁿ⁾=±F/⁽ⁿ⁻²⁾Γ⁰Γ^p, givingσ1±Γ⁰Γ^pwhich removes terms due to ourκgauge choice. In this particular setting the contributions to ˆgand ˆBtake the same overall form. Nevertheless the lacking factor ofiremains a problem.

since Γ⁰Γ^p commutes with transverse Γ matrices but anti-commutes with light cone ones, and we have ¯θΓ⁰Γ^p = −θ. This means that everything is precisely compatible, provided¯ the forms of the mirror background are given by

S˜=−iΓ⁰Γ^pSˇ=−ibcΓ⁰Γ^pS

Γ^0/5→iΓ^5/0 σ1→−σ₁

. (4.30)

We can actually rewrite this very nicely. Since S must have one and only one light cone index, we can always write it as Γ⁰N + Γ^pK for someN and K that do not contain Γ⁰ or Γ^p. In this form it is clear however that multiplication by Γ⁰Γ^p undoes the interchange of Γ⁰ for Γ^p in ˇS, leaving just an extra factor ofi. In short, we have

S˜=−iΓ⁰Γ^pSˇ=bcS

σ1→−σ₁. (4.31)

This transformation is uniquely fixed up to the sign choice bc = ±1. However, since a simultaneous sign change on all RR fields leaves the supergravity equations of motion [35]

invariant, we can consider this sign choice irrelevant anyway.

We see that two backgrounds are related by a double Wick rotation provided that the metric and B field are related as in figure 1 and the dilaton and RR fields as

e^ΦF/⁽ⁿ⁾→(i)ⁿ⁻¹e^ΦF/⁽ⁿ⁾, (4.32) where we have chosen bc = 1. With this choice, the story unifies at the level of the underlying even degree forms (see appendix A.4)

e^Φd /C⁽ⁿ⁾→iⁿe^Φd /C⁽ⁿ⁾, d /B→i²d /B.

(4.33) In short, up to a possible sign the combinationse^ΦF/ for the mirror background are identical to the originals. Of course the slashes can be removed, but then it is important to note that the relations hold between tensors with flat space indices. Our full story can now be summarized by figure 3. At this level we cannot immediately disentangle the dilaton and the RR fields.

5 Double Wick related string backgrounds

In the previous section we found transformation laws for the background fields via an explicit double Wick rotation on the world sheet combined with an appropriate fermionic field redefinition. Before giving a few examples of pairs of backgrounds related by these transformations, let us discuss how these transformation laws can also be viewed more directly from a target space point of view.

As used in our appendix, light cone gauge fixing can be viewed as T dualizing in thex⁻ direction, and gauge fixing the corresponding T dual fieldψ=σin addition tox⁺=τ [36].

It should then be possible to (formally) view a double Wick rotation on the world sheet at the target space level as a T duality inx⁻, followed by the analytic continuation (x⁺, ψ)→

L L ˜

τ→i˜σ, σ→ −i˜τ

glc→ _|det^glc_g

lc|, B→ −B, e^ΦF/⁽ⁿ⁾→(i)ⁿ⁻¹e^ΦF/⁽ⁿ⁾

Figure 3. A double Wick rotation as a change of background fields. A double Wick rotation of the light cone gauge fixed world sheet theory of a Green-Schwarz string is equivalent to a change of background fields, at least to second order in the fermions. It is not clear whether these changed fields always correspond to a solution of supergravity.

(iψ,˜ −i˜x⁺), and finally another T duality in the ˜ψ direction.¹⁵ As this procedure involves analytic continuation of target space fields as well as T duality in a light cone direction however,¹⁶ this procedure takes us out of the realm of real supergravity and string theory, and generically does not result in a real solution of supergravity. Furthermore, the details of this analytic continuation are subtle, as the complex ‘diffeomorphism’ corresponding to the double Wick rotation is ‘improper’, i.e.

τ˜(˜x⁺) σ( ˜˜ ψ)

!

= 0 i

−i0

! τ(x⁺) σ(ψ)

!

(5.1) is a transformation with determinant−1. From this point of view our transformation laws should be compatible with the supergravity equations of motion as follows.

Firstly we observe that precisely under our restrictions on the metric and B field, and the ones just found for the RR fields, the combination of T dualities and analytic continuation preserves reality of the background fields. Namely, T dualizing metrics and B fields that fit our restrictions cannot lead to mixing of light cone and transverse directions, which are the components that would pick up factors of iunder the analytic continuation.

Similarly, given RR fields with a single light cone index, T duality in x⁻ results in (type IIA) RR fields with either no ψ or + index or both, which hence remain real under the analytic continuation.

This sequence of T duality, analytic continuation, and T duality, should match the transformations we found based on our world sheet perspective. To avoid technical com- plications, we will demonstrate this explicitly for metrics diagonal in t and φ, and fixing the static gauget=τ,p_φ= 1.¹⁷ By assumption our RR fields then only have components

15We would like to thank A. Tseytlin for this suggestion.

16In principle this light cone T duality can be avoided by considering the static gauge instead of our generalized light cone gauge, leading to T dualities in theφdirections.

17This gauge is equivalent to our light cone gauge fora= 0.

involvingtorφ, and upon T dualizing φtoψ we get F_t...

Fφ...

!

T duality

−→ F_tψ...

−F_...

!

, (5.2)

where the dots denote an even number of transverse indices. The analytic continuation is slightly subtle however, involving more than the simple replacement (t, ψ) →(iψ,˜ −i˜t).

The reason for this can be seen from the bosonic type IIA supergravity action (see e.g.

page 172 of [37]). While this action is generically diffeomorphism invariant, this involves a little more work for improper diffeomorphisms such as ours, due to the Chern-Simons term B∧F⁽⁴⁾∧F⁽⁴⁾. This term behaves like a pseudoscalar under diffeomorphisms, and picks up a sign under our analytic continuation.¹⁸ We can fix this sign by combining our analytic continuation with a sign flip onB. However, to then keep the kinetic term|Fˆ⁽⁴⁾|² invariant, since ˆF⁽⁴⁾ =F⁽⁴⁾−C⁽¹⁾∧H, we also need to flip the sign of eitherF⁽⁴⁾ orF⁽²⁾ (C⁽¹⁾). Importantly, this introduces a relative sign between F⁽ⁿ⁾ and F⁽ⁿ⁺²⁾. The result of the analytic continuation and a second T duality then becomes

F_tψ...

−F_...

!

an. cont.

−→ ± F_˜tψ...˜

−F_...

!

T duality

−→ ± F˜t...

Fφ...˜

!

, (5.3)

where in the analytic continuation step we implicitly swapped the indices for a sign, and the overall±sign refers to the degree of the form, being plus forF⁽¹⁾ and F⁽⁵⁾, and minus for F⁽³⁾, or vice versa.¹⁹ Combining this with the sign flip ofBunder the analytic continuation, and the dilaton generated by the double T duality that is clearly compatible with e^ΦF/ = e^Φ˜F, we see that this precisely matches the transformations we found previously from a˜/ world sheet perspective, which are therefore compatible with the supergravity equations of motion.

After this general discussion let us give some examples. While our restrictions on the metric and B field do not appear too drastic, the number of explicitly known solutions of supergravity is also not very large; we will consider AdS₅×S⁵, AdS₃×S³×T⁴ supported by a RR three form, and AdS₂×S²×T⁶.²⁰ We distinguish mirror space quantities by tildes below.

AdS₅×S⁵

The metric of AdS₅×S⁵ in global coordinates ds²= −(1 +ρ²)dt²+ dρ²

1 +ρ² +ρ²dΩ₃+ (1−r²)dφ²+ dr²

1−r² +r²dΩ₃, (5.4)

18To see this explicitly, note that our RR fields T dualized to type IIA have either both t and ψ as components, or neither, where the contribution with both picks up a sign under analytic continuation.

19Recall that we encountered the same inconsequential sign freedom in our world sheet discussion above.

20To have a case with a transverse B fields we could for example consider the three parameter general- ization of the Lunin-Maldacena background [38] constructed in [39], which fits our restrictions when only γ1 is nonzero. Presumably the associated mirror space can be obtained by TsT transformations on mirror AdS5×S⁵; we will not pursue this interesting case here.

is precisely of the form discussed in section 2, and its mirror companion is [14]

ds˜² = − 1

1−r²dt²+ dρ²

1 +ρ² +ρ²dΩ₃+ 1

1 +ρ²dφ²+ dr²

1−r² +r²dΩ₃. (5.5) Note that this metric corresponds to a direct product of two manifolds with coordinates t, r, and associated angles, and φ, ρ, and associated angles. We have written the mirror metric with this nonstandard ordering of coordinates, so that we leave the labeling of the transverse space untouched with respect to AdS₅×S⁵.

Both these spaces can be embedded in type IIB supergravity, supported by a self dual five form and dilaton. The relevant equations of motion are given in appendix A.4. For AdS₅×S⁵ these equations are solved by a constant dilaton Φ = Φ₀, and a five form

1

4e^ΦF/ = Γ⁰¹²³⁴−Γ⁵⁶⁷⁸⁹. (5.6)

In line with the above discussion, mirror AdS₅×S⁵ [14] has a nontrivial dilaton Φ = ˜˜ Φ₀−1

2log(1−r²)(1 +ρ²), (5.7) and its combination with the five form is precisely such that

1

4e^Φ˜F˜/ = Γ⁰¹²³⁴−Γ⁵⁶⁷⁸⁹= ¹₄e^ΦF ./ (5.8) Note that since the metrics differ, the physical meaning of the forms is very different between AdS₅×S⁵ and its mirror version; in the mirror case the form mixes directions belonging to the two five dimensional submanifolds, while the form is just a difference of five dimensional volume forms for AdS₅×S⁵.

AdS₃×S³×T⁴ and AdS₂×S²×T⁶

Taking our φ coordinate to be the equatorial angle on the sphere, the mirror metrics associated to AdS₃×S³×T⁴ and AdS₂×S²×T⁶ are just the lower dimensional analogues of eq. (5.5) completed to ten dimensions with flat directions. We choose to label coordinates such thatφ∼n, hencep=n, for the AdS_n×Sⁿ case.

The supergravity equations of motion are solved by a constant dilaton for AdS₃×S³×T⁴ and AdS₂×S²×T⁶. We take AdS₃×S³×T⁴ to be supported by the three form

1

2e^ΦF/ = Γ⁰¹²+ Γ³⁴⁵, (5.9)

while AdS₂×S²×T⁶ is supported by the five form e^ΦF/ = Γ⁰¹Re(w/

C³)−Γ²³Im(w/

C³), (5.10)

where w_C³ is the holomorphic volume form on T⁶ in complex coordinates. Based on our discussion above, this should then also give the solution for their mirror versions. Indeed, we can readily check that this is the case with the mirror space dilaton in both cases given by eq. (5.7).²¹

21These mirror space solutions can be independently derived by making an obvious ansatz based on the result for AdS5×S⁵. Alternately, they can also be obtained by T dualizing intandφ, giving dS2/3×H^2/3× T^6/4 which are simple solutions of type IIB^∗supergravity analogous to AdS2/3×S^2/3×T^6/4 in type IIB supergravity.